Question

The projection of $2\hat{i}-3\hat{j}+4\hat{k}$ on the line whose equation is $\overset{ \rightarrow }{r}=\left(3 + \lambda \right)\hat{i}+\left(3 - 2 \lambda \right)\hat{j}+\left(5 + 6 \lambda \right)\hat{k}$ , where $\lambda $ is a scalar parameter, is

• A. $\frac{6}{\sqrt{41}}$
• B. $\frac{32}{\sqrt{41}}$
• C. $\frac{16}{\sqrt{41}}$
• D. $\frac{7}{5}$

Answer 1

Answer: (B)

Given line is parallel to $\hat{i}-2\hat{j}+6\hat{k}=\overset{ \rightarrow }{b}$ (let)
Let $\overset{ \rightarrow }{a}=2\hat{i}-3\hat{j}+4\hat{k}$
Projection of $\overset{ \rightarrow }{a}$ on $\overset{ \rightarrow }{b}$ is $\frac{\overset{ \rightarrow }{a} \cdot \overset{ \rightarrow }{b}}{\left|\overset{ \rightarrow }{b}\right|} = \frac{2 + 6 + 24}{\sqrt{41}} = \frac{32}{\sqrt{41}}$